BdMO 2013 Higher Secondary Problem 9
 FahimFerdous
 Posts: 176
 Joined: Thu Dec 09, 2010 12:50 am
 Location: Mymensingh, Bangladesh
BdMO 2013 Higher Secondary Problem 9
Six points $A, B, C, D, E, F$ are chosen on a circle anticlockwise. None of $AD, BE, CF$ is a diametre. Extended $AB$ and $DC$ meet at $Z$, $CD$ and $FE$ at $X$, $EF$ and $BA$ at $Y$. $AC$ and $BF$ meets at $P$, $CE$ and $BD$ at $Q$ and $AE$ and $DF$ at $R$. If $O$ is the point of intersection of $YQ$ and $ZR$, find $\angle XOP$.
Your hot head might dominate your good heart!
 FahimFerdous
 Posts: 176
 Joined: Thu Dec 09, 2010 12:50 am
 Location: Mymensingh, Bangladesh
Re: BdMO 2013 Higher Secondary Problem 9
Let $AD \cap EF=T$, $BE \cap CD=S$, $AB \cap CF=K$, $AE \cap BD=L$, $CR \cap FQ=M$. Then use Pascal's theorem and Desaurge's theorem to prove that $P, Q, T; P, R, S; R, Q, K; X, O, M; L, M, P; X, L, P$ are collinear (I'm leaving this tedious job to you guys, I can't write it). And it implies that $X, O, P$ are collinear.
I still can't believe I missed this and problem 8.
I still can't believe I missed this and problem 8.
Your hot head might dominate your good heart!

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Re: BdMO 2013 Higher Secondary Problem 9
You spin my head right round right round,
When you go down, when you go down down......($from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......($from$ "$THE$ $UGLY$ $TRUTH$" )
Re: BdMO 2013 Higher Secondary Problem 9
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Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah  Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah  Mahi